Is there a way to say the equivalent of "the day", as in
"The day you learn to speak Hebrew will be the day I buy you a new car"
"That will be the day"
"We will be in Italy the day that you leave"
"hayom" is simply ambiguous between 'today' and 'the day'. it's equally good in both meanings (though the the first one is more common).
however, in some cases, like in the example that you gave "we will be in italy the day that you leave", you should say "bayom", meaning 'on the day' (or 'in' or 'at', i'm not sure about the correct preposition in english).
by the way, i would appreciate corrections of my english...
Also, you could use כיום, which also means "today". Although some people might take it to mean "like/according to/around today", because כ can mean "like/according to/around ("around" in the sense of "about")".
Today has two meanings: "on or for this day", "at the present time". The usual in modern Hebrew is כיום for the latter, היום for the former, but in certain situations היום is also for the latter, e.g. "הנוער של היום".